Symmetry-Informed Term Filtering for Continuum Equation Discovery

TL;DR

This paper introduces an algebraic filtering approach that systematically enumerates all symmetry-allowed terms in continuum equations, facilitating data-driven discovery of physical laws with symmetry constraints.

Contribution

It presents a novel linear operator-based method to efficiently identify all symmetry-permitted terms, improving the process of formulating governing equations for complex systems.

Findings

Successfully identified invariant terms for systems with dihedral symmetry.

Recovered governing equations for Toner--Tu and Kardar--Parisi--Zhang models.

Extended known models with higher-order terms.

Abstract

Discovering governing equations, whether manually or by data-driven methods, has been central in physics and related areas. Since governing equations are typically constrained by a set of symmetries, using symmetry constraints to restrict terms is usually the first step in manually formulating a governing equation, but it often becomes intractable for complex systems with high-order derivatives or multiple fields. When a data-driven method is used, on the other hand, imposing physical constraints such as symmetries typically requires manual preprocessing or computationally expensive iterative procedures. Here, we propose an algebraic filtering method that enumerates all symmetry-allowed terms for continuum equations within a finite candidate space. By treating symmetry generators as linear operators on the candidate space, we reduce the problem of enforcing both discrete and continuous symmetries to solving a set of linear kernel equations. The solution yields a provably complete list of permitted terms. We demonstrate the method's effectiveness by identifying invariant terms for systems with dihedral symmetry and recovering the governing equations for the Toner--Tu and Kardar--Parisi--Zhang systems, including higher-order terms useful for extending known models. The method provides a systematic way to obtain a symmetry-allowed search space for data-driven equation discovery, e.g., the sparse identification of nonlinear dynamics method.